Abstract. In this paper, numerical studies for fractional differential equations (FDEs) which are... more Abstract. In this paper, numerical studies for fractional differential equations (FDEs) which are generated by optimization problem are studied using the pseudo-spectral method based upon Legendre approximations and fractional finite difference method (FDM). The fractional derivatives are presented in terms of Caputo sense. The application of the proposed methods to the generated system of FDEs leads to algebraic system which can be solved by the Newton iteration method.
Abstract: In this paper, we consider the explicit finite-difference method for solving variable o... more Abstract: In this paper, we consider the explicit finite-difference method for solving variable order fractional heat equation with a linear source term, via the Caputo concept of variable order fractional derivative. The stability of the method is discussed by means of the Gerschgorin theorem and using the stability matrix analysis. Numerical solutions of some variable order heat equation models are presented to demonstrate the effectiveness of the method.
Abstract Fractional differential equations have recently been applied in various areas of enginee... more Abstract Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bioengineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional diffusion equation (FDE) is considered. The fractional derivative is described in the Caputo sense. The method is based upon Legendre approximations.
Abstract. In this article, a numerical study for the fractional wave equations is introduced by u... more Abstract. In this article, a numerical study for the fractional wave equations is introduced by using a class of finite difference methods. These methods are extension of the weighted average methods for ordinary (non-fractional) wave equations. The stability analysis of the proposed methods is given by a recently proposed procedure similar to the standard John von Neumann stability analysis.
Abstract: In this article we consider the approximation of singularly perturbed boundary value pr... more Abstract: In this article we consider the approximation of singularly perturbed boundary value problems using a local adaptive grid h-refinement for finite element method, the variation iteration method and the homotopy perturbation method. The solution to such problems contains boundary layers which overlap and interact and the numerical approximation must take this into account in order for the resulting scheme to converge uniformly with respect to the singular perturbation parameters.
A numerical method for solving fourth-order integro-differential equations is presented. This met... more A numerical method for solving fourth-order integro-differential equations is presented. This method is based on replacement of the unknown function by a truncated series of well-known shifted Chebyshev expansion of functions. An approximate formula of the integer derivative is introduced. The introduced method converts the proposed equation by means of collocation points to system of algebraic equations with shifted Chebyshev coefficients.
Abstract: In this paper, finite difference method (FDM) and variational iteration method (VIM) ha... more Abstract: In this paper, finite difference method (FDM) and variational iteration method (VIM) have been successfully implemented for solving nonlinear fractional-order Logistic equation (FOLE). We have apply the concepts of fractional calculus to the well known population growth model in chaotic dynamic. The fractional derivative is described in the Caputo sense. The result is generalized of the classical population growth model to arbitrary order.
Abstract: We investigate two stable methods for solving the system of linear algebraic equations ... more Abstract: We investigate two stable methods for solving the system of linear algebraic equations arising from plane, singular boundary-value problems for Laplace's equation for rectangular domains [1, 21]. The dynamical systems method and the variational regularization method are applied to derive stable solutions for these systems, and the results are compared with those obtained from the QR-factorization technique contaminated with noise.
Abstract In this paper, Chebyshev spectral method is presented for solving the non-linear fractio... more Abstract In this paper, Chebyshev spectral method is presented for solving the non-linear fractional Riccati differential equation (FRDE). The fractional derivative is described in the Caputo sense. The existence and the uniqueness of the proposed problem are given. The properties of the Chebyshev polynomials are used to reduce FRDE to the solution of non-linear system of algebraic equations using Newton iteration method.
Abstract: In this paper, Crank-Nicholson method for solving fractional wave equation is considere... more Abstract: In this paper, Crank-Nicholson method for solving fractional wave equation is considered. The stability and consistency of the method are discussed by means of Greschgorin theorem and using the stability matrix analysis. Numerical solutions of some wave fractional partial differential equation models are presented. The results obtained are compared to exact solutions. Key words: Crank-Nicholson method• fractional wave equation• stability condition• stability matrix analysis• Greschgorin theorem
Numerical studies for fractional-order Logistic differential equation with two different delays
A numerical method for solving the fractional-order logistic differential equation with two diffe... more A numerical method for solving the fractional-order logistic differential equation with two different delays (FOLE) is considered. The fractional derivative is described in the Caputo sense. The proposed method is based upon Chebyshev approximations. The properties of Chebyshev polynomials are utilized to reduce FOLE to a system of algebraic equations. Special attention is given to study the convergence and the error estimate of the presented method.
A Chebyshev pseudo-spectral method for solving fractional order integro-differential equations
Abstract A Chebyshev pseudo-spectral method for solving numerically linear and nonlinear fraction... more Abstract A Chebyshev pseudo-spectral method for solving numerically linear and nonlinear fractional-order integro-differential equations of Volterra type is considered. The fractional derivative is described in the Caputo sense. The suggested method reduces these types of equations to the solution of linear or nonlinear algebraic equations. Special attention is given to study the convergence of the proposed method.
Abstract: In this paper, two different methods are studied to solve numerically time dependent Ma... more Abstract: In this paper, two different methods are studied to solve numerically time dependent Maxwell's equations in two dimensions. The methods are, the Discontinuous Galerkin Method (DGM) and the Variational Iteration Method (VIM). Discontinuous Galerkin method is tested for different number of triangulation elements, different time intervals and different orders of basis functions. Comparisons between the solutions of Maxwell's equations using DGM and VIM are presented.
Abstract In this paper, the homotopy perturbation method (HPM) is developed to solve the focusing... more Abstract In this paper, the homotopy perturbation method (HPM) is developed to solve the focusing Manakov system of coupled nonlinear Schrödinger equations in one space variable and two space variables. The advantages of this method are, fast convergent, does not require discretizations of space-time variables and does not require to solve the resultant nonlinear system of discrete equations. Numerical examples are shown to demonstrate the accuracy and capability of the method.
Abstract In this study, Legendre collocation method is presented to solve numerically the Fredhol... more Abstract In this study, Legendre collocation method is presented to solve numerically the Fredholm-Hammerstein integral equations. This method is based on replacement of the unknown function by truncated series of well known Legendre expansion of functions. The proposed method converts the equation to matrix equation, by means of collocation points on the interval [− 1, 1] which corresponding to system of algebraic equations with Legendre coefficients.
Abstract—We are concerned with an a posteriori error analysis of adaptive finite element approxim... more Abstract—We are concerned with an a posteriori error analysis of adaptive finite element approximations of boundary control problems for second order elliptic boundary value problems under bilateral bound constraints on the control which acts through a Neumann type boundary condition. In particular, the analysis of the errors in the state, the co-state, the control, and the co-control invokes an efficient and reliable residual-type a posteriori error estimator as well as data oscillations.
Abstract: In this article, the homotopy analysis method (HAM) for obtaining semi analytical solut... more Abstract: In this article, the homotopy analysis method (HAM) for obtaining semi analytical solutions of biharmonic equations is introduced. Series solutions of the problem under consideration are developed by means of HAM and the recurrence relations are given explicitly. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions.
Abstract In this article, a numerical study is introduced for the fractional reaction-subdiffusio... more Abstract In this article, a numerical study is introduced for the fractional reaction-subdiffusion equations by using an efficient class of finite difference methods. The proposed scheme is based on Hermite formula. The stability analysis and the convergence of the proposed methods are given by a recently proposed procedure similar to the standard John von Neumann stability analysis.
Abstract In this paper, finite difference method (FDM) and Pade'-variational iteration method (Pa... more Abstract In this paper, finite difference method (FDM) and Pade'-variational iteration method (Pade'-VIM) are successfully implemented for solving the nonlinear fractional Riccati differential equation. The fractional derivative is described in the Caputo sense. The existence and the uniqueness of the proposed problem are given. The resulting nonlinear system of algebraic equations from FDM is solved by using Newton iteration method; moreover the condition of convergence is verified.
An analysis of the molecular dynamics of ethanol solvated by water molecules in the absence and p... more An analysis of the molecular dynamics of ethanol solvated by water molecules in the absence and presence of the Pt (111) surface has been performed using DL_POLY version 2.19. The structure and diffusion properties of an ethanol–water system have been studied at various temperatures from 250 to 350K. We have measured the self-diffusion coefficients of a 50: 50% ethanol–water system; in the absence of a Pt surface our results have shown an excellent agreement with the experimental data (within an error of 7.4%).
Abstract. In this paper, numerical studies for fractional differential equations (FDEs) which are... more Abstract. In this paper, numerical studies for fractional differential equations (FDEs) which are generated by optimization problem are studied using the pseudo-spectral method based upon Legendre approximations and fractional finite difference method (FDM). The fractional derivatives are presented in terms of Caputo sense. The application of the proposed methods to the generated system of FDEs leads to algebraic system which can be solved by the Newton iteration method.
Abstract: In this paper, we consider the explicit finite-difference method for solving variable o... more Abstract: In this paper, we consider the explicit finite-difference method for solving variable order fractional heat equation with a linear source term, via the Caputo concept of variable order fractional derivative. The stability of the method is discussed by means of the Gerschgorin theorem and using the stability matrix analysis. Numerical solutions of some variable order heat equation models are presented to demonstrate the effectiveness of the method.
Abstract Fractional differential equations have recently been applied in various areas of enginee... more Abstract Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bioengineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional diffusion equation (FDE) is considered. The fractional derivative is described in the Caputo sense. The method is based upon Legendre approximations.
Abstract. In this article, a numerical study for the fractional wave equations is introduced by u... more Abstract. In this article, a numerical study for the fractional wave equations is introduced by using a class of finite difference methods. These methods are extension of the weighted average methods for ordinary (non-fractional) wave equations. The stability analysis of the proposed methods is given by a recently proposed procedure similar to the standard John von Neumann stability analysis.
Abstract: In this article we consider the approximation of singularly perturbed boundary value pr... more Abstract: In this article we consider the approximation of singularly perturbed boundary value problems using a local adaptive grid h-refinement for finite element method, the variation iteration method and the homotopy perturbation method. The solution to such problems contains boundary layers which overlap and interact and the numerical approximation must take this into account in order for the resulting scheme to converge uniformly with respect to the singular perturbation parameters.
A numerical method for solving fourth-order integro-differential equations is presented. This met... more A numerical method for solving fourth-order integro-differential equations is presented. This method is based on replacement of the unknown function by a truncated series of well-known shifted Chebyshev expansion of functions. An approximate formula of the integer derivative is introduced. The introduced method converts the proposed equation by means of collocation points to system of algebraic equations with shifted Chebyshev coefficients.
Abstract: In this paper, finite difference method (FDM) and variational iteration method (VIM) ha... more Abstract: In this paper, finite difference method (FDM) and variational iteration method (VIM) have been successfully implemented for solving nonlinear fractional-order Logistic equation (FOLE). We have apply the concepts of fractional calculus to the well known population growth model in chaotic dynamic. The fractional derivative is described in the Caputo sense. The result is generalized of the classical population growth model to arbitrary order.
Abstract: We investigate two stable methods for solving the system of linear algebraic equations ... more Abstract: We investigate two stable methods for solving the system of linear algebraic equations arising from plane, singular boundary-value problems for Laplace's equation for rectangular domains [1, 21]. The dynamical systems method and the variational regularization method are applied to derive stable solutions for these systems, and the results are compared with those obtained from the QR-factorization technique contaminated with noise.
Abstract In this paper, Chebyshev spectral method is presented for solving the non-linear fractio... more Abstract In this paper, Chebyshev spectral method is presented for solving the non-linear fractional Riccati differential equation (FRDE). The fractional derivative is described in the Caputo sense. The existence and the uniqueness of the proposed problem are given. The properties of the Chebyshev polynomials are used to reduce FRDE to the solution of non-linear system of algebraic equations using Newton iteration method.
Abstract: In this paper, Crank-Nicholson method for solving fractional wave equation is considere... more Abstract: In this paper, Crank-Nicholson method for solving fractional wave equation is considered. The stability and consistency of the method are discussed by means of Greschgorin theorem and using the stability matrix analysis. Numerical solutions of some wave fractional partial differential equation models are presented. The results obtained are compared to exact solutions. Key words: Crank-Nicholson method• fractional wave equation• stability condition• stability matrix analysis• Greschgorin theorem
Numerical studies for fractional-order Logistic differential equation with two different delays
A numerical method for solving the fractional-order logistic differential equation with two diffe... more A numerical method for solving the fractional-order logistic differential equation with two different delays (FOLE) is considered. The fractional derivative is described in the Caputo sense. The proposed method is based upon Chebyshev approximations. The properties of Chebyshev polynomials are utilized to reduce FOLE to a system of algebraic equations. Special attention is given to study the convergence and the error estimate of the presented method.
A Chebyshev pseudo-spectral method for solving fractional order integro-differential equations
Abstract A Chebyshev pseudo-spectral method for solving numerically linear and nonlinear fraction... more Abstract A Chebyshev pseudo-spectral method for solving numerically linear and nonlinear fractional-order integro-differential equations of Volterra type is considered. The fractional derivative is described in the Caputo sense. The suggested method reduces these types of equations to the solution of linear or nonlinear algebraic equations. Special attention is given to study the convergence of the proposed method.
Abstract: In this paper, two different methods are studied to solve numerically time dependent Ma... more Abstract: In this paper, two different methods are studied to solve numerically time dependent Maxwell's equations in two dimensions. The methods are, the Discontinuous Galerkin Method (DGM) and the Variational Iteration Method (VIM). Discontinuous Galerkin method is tested for different number of triangulation elements, different time intervals and different orders of basis functions. Comparisons between the solutions of Maxwell's equations using DGM and VIM are presented.
Abstract In this paper, the homotopy perturbation method (HPM) is developed to solve the focusing... more Abstract In this paper, the homotopy perturbation method (HPM) is developed to solve the focusing Manakov system of coupled nonlinear Schrödinger equations in one space variable and two space variables. The advantages of this method are, fast convergent, does not require discretizations of space-time variables and does not require to solve the resultant nonlinear system of discrete equations. Numerical examples are shown to demonstrate the accuracy and capability of the method.
Abstract In this study, Legendre collocation method is presented to solve numerically the Fredhol... more Abstract In this study, Legendre collocation method is presented to solve numerically the Fredholm-Hammerstein integral equations. This method is based on replacement of the unknown function by truncated series of well known Legendre expansion of functions. The proposed method converts the equation to matrix equation, by means of collocation points on the interval [− 1, 1] which corresponding to system of algebraic equations with Legendre coefficients.
Abstract—We are concerned with an a posteriori error analysis of adaptive finite element approxim... more Abstract—We are concerned with an a posteriori error analysis of adaptive finite element approximations of boundary control problems for second order elliptic boundary value problems under bilateral bound constraints on the control which acts through a Neumann type boundary condition. In particular, the analysis of the errors in the state, the co-state, the control, and the co-control invokes an efficient and reliable residual-type a posteriori error estimator as well as data oscillations.
Abstract: In this article, the homotopy analysis method (HAM) for obtaining semi analytical solut... more Abstract: In this article, the homotopy analysis method (HAM) for obtaining semi analytical solutions of biharmonic equations is introduced. Series solutions of the problem under consideration are developed by means of HAM and the recurrence relations are given explicitly. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions.
Abstract In this article, a numerical study is introduced for the fractional reaction-subdiffusio... more Abstract In this article, a numerical study is introduced for the fractional reaction-subdiffusion equations by using an efficient class of finite difference methods. The proposed scheme is based on Hermite formula. The stability analysis and the convergence of the proposed methods are given by a recently proposed procedure similar to the standard John von Neumann stability analysis.
Abstract In this paper, finite difference method (FDM) and Pade'-variational iteration method (Pa... more Abstract In this paper, finite difference method (FDM) and Pade'-variational iteration method (Pade'-VIM) are successfully implemented for solving the nonlinear fractional Riccati differential equation. The fractional derivative is described in the Caputo sense. The existence and the uniqueness of the proposed problem are given. The resulting nonlinear system of algebraic equations from FDM is solved by using Newton iteration method; moreover the condition of convergence is verified.
An analysis of the molecular dynamics of ethanol solvated by water molecules in the absence and p... more An analysis of the molecular dynamics of ethanol solvated by water molecules in the absence and presence of the Pt (111) surface has been performed using DL_POLY version 2.19. The structure and diffusion properties of an ethanol–water system have been studied at various temperatures from 250 to 350K. We have measured the self-diffusion coefficients of a 50: 50% ethanol–water system; in the absence of a Pt surface our results have shown an excellent agreement with the experimental data (within an error of 7.4%).
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Papers by Nasser Sweilam